Distinct clock speeds give identical full pointer records on the shell
Revealing clock position and receiver energy permits local recovery, but it does not give recovery on the full receiver energy shell for an admissible fixed pulse design. Two distinct initial clock speeds can produce identical eight final pointer records from distinct receiver states of the same energy. The construction persists at positive coupling and separates both canonical coordinates, giving a preparation-dependent reconstruction-error product.
R22, 2026-09-11. Use R20’s early-pulse design and R21’s revealed offset s_0, known zero incoming probe momenta, exact energy E and known observation time T. Initial speed ranges over a fixed positive interval about v_0; incoming positions occupy a fixed positive box. Receiver support is the entire shell H_s=E. Choose the design sufficiently early as below, then hold all pulse parameters fixed. Source coverage is in B53.
1. The early-pulse speed derivative is indefinite in energy
Let O have rows e_x M^n, n=0,1,2,3, and let B_epsilon=-A_{v_0}^{-1}partial_v A_{v_0}. The moment expansion used in R20 gives, in fixed component units,
\[B_\varepsilon=B^{(0)}+O(\varepsilon),\qquad B^{(0)}=\frac1{v_0}O^{-1}\operatorname{diag}(1,2,3,4)O. \tag{1}\]
To check the inverse order, write A=K r epsilon W D O plus its analytic Taylor tail, with D=diag(1,r epsilon,(r epsilon)^2,(r epsilon)^3). After left multiplication by D inverse W inverse and removal of K r epsilon, the tail and its v derivative are O(epsilon). The leading derivative gives the displayed diagonal weights; all matrices and their inverses here use the fixed nonzero moment matrix W. This proves (1) without inverting a bare vanishing determinant estimate.
The four initial output derivatives in R06 translate (1) into
\[v_0B^{(0)}z= \left(x,\ 2P,\ 3y-\frac{2a}{g}x, 4Q-\frac{2a\nu}{g\mu}P\right). \tag{2}\]
For Q=0, using a=3g and d=5g/3, the energy-speed quadratic form is
\[v_0 z^TGB^{(0)}z =g(9x^2-14xy+5y^2)+\frac{2P^2}{\mu}. \tag{3}\]
In particular the potential term is g(x-y)(9x-5y), with both signs. Choose a physical length L>0 and momentum p>0 with 2p2/mu<4gL2/5. The vectors (L,p,0,0) and (L,p,7L/5,0) have positive and negative values respectively. Normalize each vector, and the path between them, onto H_s=E:
\[z(r)=\sqrt{\frac{E}{H_s(L,p,rL,0)}}(L,p,rL,0), \qquad 0\le r\le7/5. \tag{4}\]
Along this compact path x and P are uniformly positive. For sufficiently small fixed epsilon, endpoint signs in (3) persist for B_epsilon, and both (B_epsilon z(r))_x and (B_epsilon z(r))_P stay uniformly positive.
2. Actual equal-energy records at two fixed distinct speeds
Fix a small speed difference d_v>0 within the preparation interval and set v_1=v_0+d_v. At scaled zero coupling identical records require
\[z'(r)=A_{v_1}^{-1}A_{v_0}z(r),\qquad q'=q=0.\]
Taylor expansion at fixed pulse design gives uniformly in r
\[z'(r)-z(r)=d_v B_\varepsilon z(r)+O(d_v^2),\] \[H_s(z'(r))-E=d_v z(r)^TGB_\varepsilon z(r)+O(d_v^2). \tag{5}\]
Choose d_v sufficiently small and then keep it fixed. The second expression has opposite signs at the two endpoints of (4), while the first has both canonical components at least c_x d_v and c_P d_v for fixed c_x,c_P>0. The intermediate value theorem gives a point r_* where both receivers have energy E and all scaled records agree, with both coordinates distinct. This establishes an actual pair rather than inferring one from a zero derivative. Constants c_x,c_P have units respectively time and mass.
3. The equal-record branches persist with full back-reaction
For each r, use the exact positive-coupling record generated by (z(r),q=0,v_0). At speed v_1 solve
\[F_{\lambda,v_1}(z'_\lambda(r),q'_\lambda(r)) =F_{\lambda,v_0}(z(r),0). \tag{6}\]
At lambda=0 its derivative in (z’,q’) is diag(-A_{v_1},I), independent of r and invertible. A common small tube around the compact path of zero-coupling solutions gives a parameter contraction, using this same linear inverse. Thus a smooth solution exists uniformly for all r, with
\[z'_\lambda(r)=z'(r)+O(\lambda),\qquad q'_\lambda(r)=O(\lambda). \tag{7}\]
All receiver states used to solve (6) lie in a fixed convex neighbourhood of the energy ball with strict cutoff margins. Shrink d_v to fit that tube, then lambda_0 to preserve it and half the positive incoming-position margins. No measured clock momentum is added. Both initial clock positions equal s_0.
Since d_v is fixed, the opposite endpoint energy signs in (5) persist for small lambda. Continuity of H_s(z’_lambda(r))-E supplies r_lambda with energy exactly E for both receivers. Reduce lambda_0 further so that the canonical separations stay at least c_x d_v/2 and c_P d_v/2. Equality (6) means equality of all eight unscaled final records for positive coupling. The speeds stay at their two fixed interior values, the first incoming position is zero and the second stays inside its fixed box. This yields admissible pairs in the full Cartesian shell/preparation support.
4. Canonical risk and its preparation dependence
Common-record pairs imply for every deterministic receiver estimator
\[\epsilon_x\ge\frac{c_xd_v}{4},\qquad \epsilon_P\ge\frac{c_Pd_v}{4},\qquad \mathcal H_{\rm rec}\ge\frac{c_xc_Pd_v^2}{16}>0. \tag{8}\]
The product has action units ML^2/T and no 2 pi normalization. These constants are uniform for sufficiently small positive coupling at fixed design, speed separation and positive position width. The bound closes as the permitted speed separation contracts. It asserts a pair and coordinate risks, not a positive canonical area or a universal action constant.
R21’s inverse remains valid on its selected local patch. The new pairs show why membership in that patch is substantive supplied information. The full shell includes different energy-response signs, permitting another receiver branch at a different speed.
Next R23: reveal the final clock momentum, which is persistent after the pulses, instead of revealing its initial value. Test whether this additional record distinguishes the two speed branches uniformly on the full shell. This asks whether a physical stored clock record replaces the local-patch prior in R21.